5.3.1: Resources and Key Concepts

Resources

Videos

• Power function
  • Intermediate Algebra - Integer Exponents: Power Functions
• End behavior
  • Power Functions and Polynomial Functions - Identifying End Behavior of Polynomial Functions
  • Polynomial Functions and Their Graphs - End Behavior of Polynomials
• Polynomial function
  • Polynomial Functions - The Concept
• Zeros (of a polynomial)
  • Graphs of Polynomial Functions - Using Factoring to Find Zeros of Polynomial Functions

Key Concepts

Definitions

• Power Function: A function that can be represented in the form \(f(x) = kx^p\), where \(k\) and \(p\) are real numbers, and \(k\) is known as the coefficient.
• End Behavior of a Function: The way the graph of the function behaves as the input (typically the \(x\)-value) approaches positive or negative infinity (denoted \(x \rightarrow \infty\) or \(x \rightarrow -\infty\)).
• Polynomial Function: A function that can be written in the form \(f(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0\), where \(n \in \mathbb{N}\) (a non-negative integer) and the coefficients \(a_i\) are real numbers.
• General Form of a Polynomial Function: The form \(f(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0\), typically with terms arranged in descending order of power.
• Coefficient (of a term in a polynomial): The real number factor multiplying the variable part of a term (e.g., in \(a_i x^i\), \(a_i\) is the coefficient).
• Term (of a polynomial function): Each product \(a_i x^i\) in the polynomial function.
• Degree (of a polynomial function): The largest exponent \(n\) for which the coefficient \(a_n\) is not zero.
• Leading Term (of a polynomial function): The term containing the highest power of the variable (the term with the highest degree).
• Leading Coefficient (of a polynomial function): The coefficient of the leading term.
• Turning Point (of a polynomial graph): A point at which the function values change from increasing to decreasing or decreasing to increasing.
• Continuous Function (informal definition): A function whose graph has no breaks; it can be drawn without lifting the pen from the paper.
• Smooth Curve (informal definition): A graph that has no sharp corners.

Theorems

• End Behavior of a Polynomial Function: Let \(P(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_0\), where \(a_n \neq 0\). Then the end behavior of the graph of \(P(x)\) will match the end behavior of the graph of its leading term \(y = a_n x^n\).
• Number of Turning Points for a Polynomial Function: A polynomial of degree \(n\) will have, at most, \(n-1\) turning points and \(n\) horizontal intercepts (zeros).

Common Mistakes

• Confusing Power Functions with Exponential Functions: A power function has a variable base and a fixed power (e.g., \(x^2\)), while an exponential function has a fixed base and a variable power (e.g., \(2^x\)). The function \(f(x)=2^x\) is not a power function.
• Incorrectly Identifying the Degree or Leading Coefficient: When a polynomial is not written in standard general form (descending powers), care must be taken to identify the term with the highest power to determine the degree and leading coefficient.
• Errors in Determining End Behavior: Misinterpreting the combined effect of the sign of the leading coefficient and the parity (even/odd) of the degree of the polynomial on its end behavior.
• Assuming Maximum Number of Turning Points/Intercepts is Always Achieved: A polynomial of degree \(n\) has *at most* \(n-1\) turning points and *at most* \(n\) \(x\)-intercepts. It does not necessarily have exactly that many.